Singular 3-manifolds in $\mathbb{R}^5$

TL;DR

This paper investigates 3-manifolds with singularities in five-dimensional space, introducing a curvature locus that captures local second-order geometric information at singular points.

Contribution

It defines the curvature locus at singular points of 3-manifolds in , providing a new tool for understanding their local geometry.

Findings

Curvature locus encodes second-order geometry at singularities

Provides a framework for analyzing singular 3-manifolds in 

Advances understanding of geometric properties near singular points

Abstract

We study 3-manifolds in $\mathbb{R}^5$ with corank $1$ singularities. At the singular point we define the curvature locus using the first and second fundamental forms, which contains all the local second order geometrical information about the manifold.